Continuous time markov chain problem from Essentials of Stochastic Processes

Question

Problem

I'm stuck on this question, I don't even know where to start here. What is the rate of the prof writing letter when there are less than k requests? Any help is appreciated.

Answer 1

Instead of a single stressed professor processing a letter with rate $2k$, where $k$ is the number of letters that have yet to be processed, you can instead look at the setup as consisting of a team of infinitely many professors, each of them capable of processing a letter with rate $2$. From this point of view, it becomes apparent that you're dealing with an $M / G / \infty$ queueing system. Although the service distribution is never said to be Poisson, the word 'rate' nonetheless makes me suspect that we're in fact looking at an $M/M/\infty$ queue with arrival rate $8$, and service rate $2$.

With this observation, questions (a) and (c) can be tackled with standard tools that I presume you have available.

Unless I misunderstand, with (b) you have to be a bit more careful, because in the translation to the $M/G/\infty$ queue the first-come-first-serve basis gets lost. Question (b), in the language of $M / G / \infty$ queues, becomes the following. Suppose a customer arrives in the $M / G / \infty$ queue and he is the $n$-th customer into the system. Given, then, that there are $n$ customers in the system, what is the expected amount of time for $n$ customers to get served by the system?

Hope this helps. I haven't fact-checked my claims so let me know if something feels off.